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# On Axiomatizability of Non-Commutative $L_p$-Spaces

Published:2007-12-01
Printed: Dec 2007
• C. Ward Henson
• Yves Raynaud
• Andrew Rizzo
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## Abstract

It is shown that Schatten $p$-classes of operators between Hilbert spaces of different (infinite) dimensions have ultrapowers which are (completely) isometric to non-commutative $L_p$-spaces. On the other hand, these Schatten classes are not themselves isomorphic to non-commutative $L_p$ spaces. As a consequence, the class of non-commutative $L_p$-spaces is not axiomatizable in the first-order language developed by Henson and Iovino for normed space structures, neither in the signature of Banach spaces, nor in that of operator spaces. Other examples of the same phenomenon are presented that belong to the class of corners of non-commutative $L_p$-spaces. For $p=1$ this last class, which is the same as the class of preduals of ternary rings of operators, is itself axiomatizable in the signature of operator spaces.
 MSC Classifications: 46L52 - Noncommutative function spaces 03C65 - Models of other mathematical theories 46B20 - Geometry and structure of normed linear spaces 46L07 - Operator spaces and completely bounded maps [See also 47L25] 46M07 - Ultraproducts [See also 46B08, 46S20]

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